Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, z\neq 0$. $\dfrac{{(p^{5})^{-1}}}{{(p^{-5}z)^{4}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${p^{5}}$ to the exponent ${-1}$ . Now ${5 \times -1 = -5}$ , so ${(p^{5})^{-1} = p^{-5}}$ In the denominator, we can use the distributive property of exponents. ${(p^{-5}z)^{4} = (p^{-5})^{4}(z)^{4}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(p^{5})^{-1}}}{{(p^{-5}z)^{4}}} = \dfrac{{p^{-5}}}{{p^{-20}z^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-5}}}{{p^{-20}z^{4}}} = \dfrac{{p^{-5}}}{{p^{-20}}} \cdot \dfrac{{1}}{{z^{4}}} = p^{{-5} - {(-20)}} \cdot z^{- {4}} = p^{15}z^{-4}$.